### 1: Congruence

#### 1.1: Experiment with transformations in the plane.

KY.HS.G.1: Know and apply precise definitions of the language of Geometry:

KY.HS.G.1.a: Understand properties of line segments, angles and circle.

KY.HS.G.1.b: Understand properties of and differences between perpendicular and parallel lines.

KY.HS.G.2: Representing transformations in the plane.

KY.HS.G.2.a: Describe transformations as functions that take points in the plane as inputs and give other points as outputs.

KY.HS.G.2.b: Compare transformations that preserve distance and angle measures to those that do not.

KY.HS.G.2.c: Given a rectangle, parallelogram, trapezoid, or regular polygon, formally describe the rotations and reflections that carry it onto itself, using properties of these figures.

KY.HS.G.3: Develop formal definitions of rotations, reflections and translations in terms of angles, circles, perpendicular lines, parallel lines and line segments.

KY.HS.G.4: Understand the effects of transformations of geometric figures.

KY.HS.G.4.a: Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure.

KY.HS.G.4.b: Specify a sequence of transformations that will carry a given figure onto another.

KY.HS.G.4.c: Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure. Given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent.

#### 1.2: Understand congruence in terms of rigid motions.

KY.HS.G.5: Know and apply the concepts of triangle congruence:

KY.HS.G.5.a: Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent.

KY.HS.G.5.b: Explain how the criteria for triangle congruence (ASA, SAS and SSS) follow from the definition of congruence in terms of rigid motions.

#### 1.3: Prove geometric theorems.

KY.HS.G.6: Apply theorems for lines, angles, triangles, parallelograms.

KY.HS.G.7: Prove theorems about geometric figures.

KY.HS.G.7.a: Construct formal proofs to justify theorems for lines, angles and triangles.

KY.HS.G.7.b: Construct formal proofs to justify theorems for parallelograms.

#### 1.4: Make geometric constructions.

KY.HS.G.8: Create and apply geometric constructions.

KY.HS.G.8.a: Make formal geometric constructions with a variety of tools and methods.

KY.HS.G.8.b: Apply basic construction procedures to construct more complex figures.

### 2: Similarity, Right Triangles and Trigonometry

#### 2.1: Understand similarity in terms of similarity transformations.

KY.HS.G.9: Understand properties of dilations.

KY.HS.G.9.a: Verify the properties that result from that dilations given by a center and a scale factor.

KY.HS.G.9.b: Verify that a dilation produces an image that is similar to the pre-image.

KY.HS.G.10: Apply the properties of similarity transformations to establish the AA criterion for two triangles to be similar.

#### 2.2: Prove theorems involving similarity.

KY.HS.G.11.c: Use similarity criteria for triangles to solve problems and to prove relationships in geometric figures.

#### 2.3: Define trigonometric ratios and solve problems involving right triangles.

KY.HS.G.12: Understand properties of right triangles.

KY.HS.G.12.a: Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles (sine, cosine and tangent).

KY.HS.G.12.b: Explain and use the relationship between the sine and cosine of complementary angles.

KY.HS.G.12.c: Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.

### 3: Circles

#### 3.1: Understand and apply theorems about circles.

KY.HS.G.15: Verify using dilations that all circles are similar.

KY.HS.G.16: Identify and describe relationships among angles and segments within the context of circles involving:

KY.HS.G.16.a: Recognize differences between and properties of inscribed, central and circumscribed angles.

KY.HS.G.16.b: Understand relationships between inscribed angles and the diameter of a circle.

KY.HS.G.17: Apply basic construction procedures within the context of a circle.

KY.HS.G.17.a: Construct the inscribed and circumscribed circles of a triangle.

#### 3.2: Find arc lengths and areas of sectors of circles.

KY.HS.G.18: Understand the relationship between an intercepted arc length within a circle and the radius of the circle.

KY.HS.G.18.b: Define the radian measure of the angle as the measure of a central angle that intercepts an arc equal in length to the radius of the circle.

### 4: Expressing Geometric Properties with Equations

#### 4.1: Translate between the geometric description and the equation for a conic section.

KY.HS.G.19: Understand the relationship between the algebraic form and the geometric representation of a circle.

KY.HS.G.19.a: Write the equation of a circle of given center and radius using the Pythagorean Theorem.

KY.HS.G.19.b: Derive and write the equation of a circle of given center and radius using the Pythagorean Theorem.

KY.HS.G.19.c: Complete the square to find the center and radius of a circle given by an equation.

KY.HS.G.20: Derive the equations of conic sections.

KY.HS.G.20.a: Derive the equation of a parabola given a focus and directrix.

KY.HS.G.20.b: Derive the equations of ellipses and hyperbolas given the foci, using the fact that the sum or difference of distances from the foci is constant.

#### 4.2: Use coordinates to prove simple geometric theorems algebraically.

KY.HS.G.22: Justify and apply the slope criteria for parallel and perpendicular lines and use them to solve geometric problems.

KY.HS.G.23: Find measurements among points within the coordinate plane.

KY.HS.G.23.a: Use points from the coordinate plane to find the coordinates of a midpoint of a line segment and the distance between the endpoints of a line segment.

KY.HS.G.24: Use coordinates within the coordinate plane to calculate measurements of two dimensional figures.

KY.HS.G.24.a: Compute the perimeters of various polygons.

### 5: Geometric Measurement and Dimensions

#### 5.1: Explain volume formulas and use them to solve problems.

KY.HS.G.25: Analyze and determine the validity of arguments for the formulas for the various figures and shapes.

KY.HS.G.25.a: Finding the circumference and area of a circle.

KY.HS.G.25.b: Finding the volume of a sphere, prism, cylinder, pyramid and cone.

KY.HS.G.26: Give an informal argument using Cavalieri’s principle for the formulas for the volume of a sphere and other solid figures.

KY.HS.G.27: Use volume formulas to solve problems for cylinders, pyramids, cones, spheres, prisms.

#### 5.2: Visualize relationships between two-dimensional and three-dimensional objects.

KY.HS.G.28: Identify the shapes of two-dimensional cross-sections of three-dimensional objects and identify three-dimensional objects generated by rotations of two-dimensional objects.

### 6: Modeling with Geometry

#### 6.1: Apply geometric concepts in modeling situations.

KY.HS.G.29: Use geometric shapes, their measures and their properties to describe objects in real world settings.

KY.HS.G.30: Apply concepts of density based on area and volume in modeling situations, using appropriate units of measurement.

Correlation last revised: 1/22/2020

This correlation lists the recommended Gizmos for this state's curriculum standards. Click any Gizmo title below for more information.